×

Submanifolds of Sasakian manifolds with certain parallel operators. (English) Zbl 1290.53058

Summary: We study submanifolds of Sasakian manifolds and obtain a condition under which certain naturally defined symmetric tensor field on the submanifold is to be parallel and use this result to obtain conditions under which a submanifold of the Sasakian manifold is an invariant submanifold.

MSC:

53C40 Global submanifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] I. Ishihara, “Anti-invariant submanifolds of a Sasakian space form,” Kodai Mathematical Journal, vol. 2, no. 2, pp. 171-186, 1979. · Zbl 0432.53034 · doi:10.2996/kmj/1138036014
[2] M. Kon, “On some invariant submanifolds of normal contact metric manifolds,” Tensor, vol. 28, pp. 133-138, 1974. · Zbl 0287.53035
[3] M. Kon, “Invariant submanifolds in Sasakian manifolds,” Mathematische Annalen, vol. 219, no. 3, pp. 277-290, 1976. · Zbl 0301.53031 · doi:10.1007/BF01354288
[4] K. Yano and M. Kon, Anti-Invariant Submanifolds, vol. 21 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1976. · Zbl 0349.53055
[5] K. Yano, M. Kon, and I. Ishihara, “Anti-invariant submanifolds with flat normal connection,” Journal of Differential Geometry, vol. 13, no. 4, pp. 577-588, 1978. · Zbl 0439.53066
[6] J. L. Cabrerizo, A. Carriazo, L. M. Fernández, and M. Fernández, “Semi-slant submanifolds of a Sasakian manifold,” Geometriae Dedicata, vol. 78, no. 2, pp. 183-199, 1999. · Zbl 0944.53028 · doi:10.1023/A:1005241320631
[7] J. L. Cabrerizo, A. Carriazo, L. M. Fernández, and M. Fernández, “Slant submanifolds in Sasakian manifolds,” Glasgow Mathematical Journal, vol. 42, no. 1, pp. 125-138, 2000. · Zbl 0957.53022 · doi:10.1017/S0017089500010156
[8] J. L. Cabrerizo, A. Carriazo, L. M. Fernández, and M. Fernández, “Structure on a slant submanifold of a contact manifold,” Indian Journal of Pure and Applied Mathematics, vol. 31, no. 7, pp. 857-864, 2000. · Zbl 0984.53034
[9] I. Hasegawa and I. Mihai, “Contact CR-warped product submanifolds in Sasakian manifolds,” Geometriae Dedicata, vol. 102, pp. 143-150, 2003. · Zbl 1066.53103 · doi:10.1023/B:GEOM.0000006582.29685.22
[10] K. Yano and M. Kon, “CR-sous-variétés d’un espace projectif complexe,” Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences. Séries A et B, vol. 288, no. 9, pp. A515-A517, 1979. · Zbl 0416.53028
[11] K. Yano and M. Kon, “Differential geometry of CR-submanifolds,” Geometriae Dedicata, vol. 10, no. 1-4, pp. 369-391, 1981. · Zbl 0464.53044 · doi:10.1007/BF01447433
[12] K. Yano and M. Kon, “Contact CR submanifolds,” Kodai Mathematical Journal, vol. 5, no. 2, pp. 238-252, 1982. · Zbl 0496.53038 · doi:10.2996/kmj/1138036553
[13] K. Yano and M. Kon, CR Submanifolds of Kaehlerian and Sasakian Manifolds, vol. 30 of Progress in Mathematics, Birkhäuser, Boston, Mass, USA, 1983. · Zbl 0496.53037
[14] A. A. Ishan, “Submanifolds of a Sasakian manifold,” Submitted. · Zbl 1324.53050
[15] D. E. Blair, Contact Manifolds in Riemannian Geometry, vol. 509 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1976. · Zbl 0319.53026 · doi:10.1007/BFb0079307
[16] B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, vol. 1 of Series in Pure Mathematics, World Scientific, Singapore, 1984. · Zbl 0537.53049
[17] K. Yano and M. Kon, “On contact CR submanifolds,” Journal of the Korean Mathematical Society, vol. 26, no. 2, pp. 231-262, 1989. · Zbl 0694.53050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.